Why Is the Definition of Injectivity Formulated This Way?

[studious]

I have a concern about the definition of injectivity:

f:U->V; f is injective, for a,b in U

1. a!=b implies f(a)!=f(b)
2. f(a)=f(b) implies a=b

Why isn't the definition:
3. a!=b if and only if f(a)!=f(b)
similarly,
4. a=b if and only if f(a)=f(b)


From 1, if a!=b implies f(a)!=f(b); consider a=b; certainly f(a)=f(b); so why isn't it that a!=b if and only if f(a)!=f(b).

What is the logic behind the definition of injectivity?

 

[HallsofIvy]

I have a concern about the definition of injectivity:

f:U->V; f is injective, for a,b in U

1. a!=b implies f(a)!=f(b)
2. f(a)=f(b) implies a=b

Why isn't the definition:
3. a!=b if and only if f(a)!=f(b)
similarly,
4. a=b if and only if f(a)=f(b)


From 1, if a!=b implies f(a)!=f(b); consider a=b; certainly f(a)=f(b); so why isn't it that a!=b if and only if f(a)!=f(b).

What is the logic behind the definition of injectivity?

Since "if a= b then f(a)= f(b)" is part of the definition of "function", it not necessary to include it in the definition of "injective function".